multi-objective design of spatial cable robots

Proceedings of the IASTED International Conference Robotics (Robo 2011) November 7 - 9, 2011 Pittsburgh, USA

MULTI-OBJECTIVE DESIGN OF SPATIAL CABLE ROBOTS Arian Bahrami1 and Mansour Nikkhah-Bahrami2 M.Sc Student, Department of Mechanical Engineering, University of Tehran,Tehran, Iran [email protected] 2 Distinguished Professors, Department of Mechanical Engineering, University of Tehran, Tehran, Iran [email protected] 1

There is some work on the optimal design of classic parallel manipulators. The isotropic configuration of stewart platform robot has been studied by a symbolic method [5]. The dexterity and stiffness of a parallel manipulator have been optimized by Gao et al. using genetic algorithms and artificial neural networks [6]. There is a difference between the isotropic design of cable robots and classical parallel robots. For cable robots, the designers must take into account the existence of tension in each cable, in addition to isotropic conditions for classical parallel robots. There are researches dealing with the optimum design of spatial cable robots. Most of the researches only focused on the workspace optimization of spatial cable array robots without considering the dexterity and stiffness indices [7, 8]. In these optimization works, one single objective function (workspace volume index) has been applied in the optimization process. Also, the optimal configuration of a 6-6 cable suspended robot has been obtained separately using the global dexterity index and workspace volume index as its objective functions [9]. In all of the mentioned optimizations, the search spaces of the design parameters have been sliced and then for every set of sliced parameters, the required objective function has been evaluated to obtain the optimal solution. This approach is quite time-consuming and the accuracy of the results depends on the size of the slices. To the best knowledge of authors, there is no complete work done on the optimization of spatial sixcable robots by means of a hybrid genetic algorithmpattern search approach considering mixed performance indices based on the combination of global dexterity index, global dexterity uniformity index, global stiffness index, global stiffness uniformity index, and workspace volume index. The organization of this paper is as follows: Section 2 describes the kinematic and design parameters of the robot. In section 3, different kinds of single-objective functions are introduced for the optimization problem. In section 4, different kinds of multi-objective functions are also introduced for the optimization problem. Section 5 presents the method used in optimization and finally, section 6 deals with the results of single and multiobjective optimization presented and interpreted through tables. The obtained results are valuable in designing these robots under different conditions.

ABSTRACT This paper presents the optimal design of spatial six-cable robots based on single and multi-objective optimization by means of a hybrid genetic algorithm-pattern search approach. The objective functions used are: Global Dexterity Index (GDI), Global Dexterity Uniformity Index (GDUI), Global Stiffness Index (GSI), and Global Stiffness Uniformity Index (GSUI). Firstly, these indices are used separately to obtain the architectural parameters of the robot. Then, mixed performance indices are specified and optimized to obtain the optimal parameters of the robot such as the size of the moving platform, the size of the fixed platform, and the geometric shapes of the two platforms. The obtained results are useful in designing these robots under different conditions.

KEY WORDS Cable robots, Dexterity, Optimal design, Stiffness, Workspace.

1. Introduction Cable array robots, also known as cable-suspended robots or cable-driven manipulators, are those robots with cables and winches. The end-effector is connected to the base platform by a number of cables. Cable array robots have several advantages compared to traditional serial or parallel robot. They have high ratio of payload to robot weight and a much larger workspace which is limited mostly by the cable lengths, appropriate for high speed applications [1], lifting heavy loads [2], camera positioning in sports stadiums [3] and many others. Some disadvantages of cable robots are cable interference and inaccuracies at the end-effector as a result of cable stretch. Based on the number of cables (m) and the number of degrees of freedom (n), there are three categories for the cable robots [4], i.e. the incompletely restrained cable robots (mn+1). For the incompletely restrained cable robots, the maximum number of cables is the same as the number of degrees of freedom of the robot. The incompletely restrained six cable robot is the main focus of this paper.

DOI: 10.2316/P.2011.752-063

345

2. Kinematic Modeling The applied model has six DOF and six cables. Such cable-array robot has a fixed platform, a moving platform and six cables each connected to a connection point on the moving platform Ai and one of the connection points on the fixed platform Bi, allowing six degrees of freedom as shown in Fig. 1. OF is considered as the origin of the fixed coordinates XYZ which coincides with the mass center of the base platform (BP) and OM is considered as the origin of the body coordinates xyz which coincides with the mass center of the moving platform (MP). rbase is the radius distance from OF to the connection points on the base platform Bi and rend is the radius distance from OM to the connection points on the moving platform Ai. Figs.2 and 3 show γ angle of BP (γbase) and MP (γend) respectively. The applied diagram of the model is shown in Fig. 4. This expression can be obtained from Fig. 4:

li  p   R  M ai  bi

i  1, 2,3,..., 6

Fig. 4. Vector polygon diagram of the robot

Where bi is the position vector of the connection points on the base platform Bi, ai is the position vector of the connection points on the moving platform Ai, p is the position vector of point OM, li is the cable length vector from Bi to Ai and R is the rotation matrix of the MP with respect to BP with three rotation angles ψ, θ, ϕ respectively about the fixed axes of X, Y and Z and can be defined as:

(1)

cos  cos   R    sin  cos   sin  

 sin  cos  cos  sin  sin cos  cos  sin  sin  sin cos sin

sin  sin  cos  sin  cos    cos sin  sin  sin  cos   cos cos 

(2) The cable length qi can be defined as:

qi  li  (liT li )1/ 2

(3)

The relation between the cable tensions and external wrench on the MP can be defined as:

Fig 1. Geometric model

J T s  Fext

(4)

Where s is the vector of cable tensions s  [ s1 s2 s3 s4 s5 s6 ]T and Fext is the vector of external wrench applied to the end-effector T Fext  [ FX FY FZ M X M Y M Z ] while FX, FY and FZ are applied forces to the MP and MX, MY and MZ are applied moments to the MP. The transpose of Jacobian matrix is written such that its ith column: Fig 2. γbase angle for BP

JTi

 li   l  i  , i  1, 2,..., 6  B li   ai   li  

(5)

Where: B

ai   R  M ai

(6)

The end-effector is deflected away from its desired position in the presence of the external forces. Increasing the robot stiffness leads to a reduction in deflection of the

Fig 3. γend angle for MP

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end-effector. The overall stiffness matrix Ku of the cable array robot can be defined as

Ku  J T diag (k1 , k2 , k3 , k4 , k5 , k6 ) J

D  diag (1,1,1,

(7) L

1 1 1 , , ) L L L

  r 2end ( x 2  y 2  2 z 2  2r 2base sin 2 (60  end  base )) 2 2( x 2  y 2  z 2   2 )

Where AE ki  i i qi

    r 2end  r 2base  2 rbase rend cos(60  end  base ) 2

(8) i  1,2,...,6

The condition number of the homogeneous Jacobian matrix can be computed at each point within the workspace volume of the robot. The global dexterity index (GDI) is a measure of kinematic dexterity of the robot over the entire workspace volume [11]. The global dexterity index (GDI) shows how far away the robot is from singularities. High values for GDI guarantee good performance with respect to force and velocity transmission [9]. It is based on the distribution of the condition number of the homogeneous Jacobian matrix over the entire workspace volume and is represented by

In which Ei is the elasticity module of the cables, Ai is the area of the cross section of the cables and qi is the cable length. This matrix is always positive semi-definite and symmetric and because of its dependence on the Jacobian, it is also dependent on the posture of the end-effector.

3. Single-Objective functions Several single-objective functions are introduced for the optimization problem. The Global Dexterity Index (GDI), Global Dexterity Uniformity Index (GDUI), Global Stiffness Index (GSI), Global Stiffness Uniformity Index (GSUI), and Workspace Volume Index (n) are the five objective functions for the optimization problem. We aim at maximizing these five objective functions in the optimization process.

 GDI 

V

(11)

V

Where V is the volume of the workspace and k is the condition number of the homogeneous Jacobian matrix. The reciprocal of the condition number ranges in value from zero to one which imposes a bound on GDI between zero and one. It is difficult to obtain the exact solution to the integral. Therefore, a discrete version of GDI can be defined as

The condition number of the Jacobian matrix is known to be a measure of the control accuracy of the robot [10]. Therefore, it is a very significant index. The minimization of this number keeps the robot away from the singularity configurations. The condition number (k) of the Jacobian matrix J can be written as

Kl Ks

1 ( )dV k dV



3.1 Dexterity and Dexterity Uniformity Indices

k(J ) 

(10)

n

 ( k1 )

GDI  d  i 1

i

(12)

n

Where n is the number of points obtained as the workspace volume of the robot and k is the condition number of the homogeneous Jacobian matrix at a particular point within the workspace volume. This index is not able to show the lower or upper bounds on the dexterity in the workspace. Therefore, another definition called Global Dexterity Uniformity Index is introduced as following

(9)

Where Kl and Ks are the maximum and the minimum singular value of the Jacobian matrix respectively. The condition number takes its values between unity and infinity. The Jacobian matrix J is well-conditioned and far away from singularities, if the condition number of J reaches 1. On the other hand as the condition number approaches infinity the matrix is ill-conditioned. As far as the Jacobian matrix is dimensionally inhomogeneous, there is a problem in computing the condition number. In Eq. (5), the first three columns of the Jacobian matrix J are dimensionless and the last three columns have the units of length. The homogeneous Jacobian matrix JH is obtained by dividing the last three columns by the characteristic length L [5], i.e., J H  JD where

 du 

min(min( i )) max(max( i ))

(13)

In which, max(max(  i )) and min(min(  i )) are the maximum and minimum singular values of the homogeneous Jacobian matrix through the entire workspace, respectively. If the Global Dexterity Uniformity Index

347

cables must be in positive tension. Therefore, at each point within the search volume if all the elements of the vector of the cable tensions obtained from below Eq. (17) have nonnegative values, that point belongs to static equilibrium workspace volume.

(GDUI) approaches to 1, the dexterity is more uniform and homogeneous through the entire workspace.

3.2 Stiffness and Stiffness Uniformity Indices Maximizing the stiffness in all directions is more appropriate rather than a particular direction [12]. Therefore, the stiffness index is a suitable measure to use in the design process. It can be computed based on the condition number of the stiffness matrix. The condition number (k) of the stiffness matrix Ku can be expressed as

k ( Ku ) 

El Es

 s  J T Fext  under si  0  i  1, 2,3,...., 6

A program was created with the use of Matlab software which tests the points within the search space to define if they belong to the static equilibrium workspace volume of the robot. By setting the values of the parameters of the robot ( rbase, rend, γbase, γend, Fx, Fy, Fz, Mx, My, Mz ), and using Eq. (17), one can obtain the number of points as the workspace volume index (n).

(14)

Where El and Es are the maximum and the minimum eigen value of the stiffness matrix respectively. The condition number takes its values between unity and infinity. The stiffness matrix is well-conditioned if the condition number reaches 1. On the other hand as the condition number approaches infinity the matrix is ill-conditioned. This quantity can also be computed at each point within the workspace volume of the robot if the the stiffness matrix Ku is homogenized as T K H  J H diag ( k1 , k 2 , k3 , k 4 , k5 , k6 ) J H to let the condition number make sense. The global stiffness index (GSI) can also be defined for all over the workspace volume in the same way as the global dexterity index (GDI). n

 GSI   s 

(

i 1

1 ) ki

4. Multi-Objective Functions In this section, multi-objective functions are considered for the optimization of the robot. The multi-objective functions based on the combination of these five global performance indices are considered as follows: 1- Overall dexterity index ODI 2- Overall stiffness index OSI 3-Overall mix of dexterity and stiffness indices ODSI We aim at maximizing these three objective functions in the optimization process. In the single-objective optimization case, we aim to obtain the best solution. But in the multi-objective optimization, objective functions can conflict with each other, so there is not necessary a best solution. The result of the multi-objective optimization is a set of solutions called Pareto-optimal solutions. By defining the weight parameters of the objective functions, one can determine one of the Pareto solutions for those specified weight parameters.

(15)

n

Where n is the number of points obtained as the workspace volume of the robot and k is the condition number of the homogeneous stiffness matrix at a particular point within the workspace volume. This index is not able to show the lower or upper bounds on the stiffness in the workspace. Therefore, another definition called Global Stiffness Uniformity Index is introduced as follows:

 su 

K min K max

(17)

4.1 Overall Dexterity Index (16) An overall dexterity index (ODI) of the manipulator can be expressed as follows:

Where Kmin and Kmax are the global minimum and maximum eigen values of the homogeneous stiffness matrix through the entire workspace, respectively [13]. If the Global Stiffness Uniformity Index (GSUI) approaches to 1, the stiffness is more uniform and homogeneous through the entire workspace.

ODI  ODI  wd1

d  du  wd 2 (d ) max ( du ) max

(18)

Where the weight parameters wd1 , wd2 (wd1 , wd2 ∈ [0 1] and wd1+wd2 =1 ) show the proportion of d (GDI),and the proportion of  du (GDUI) in the overall dexterity

3.3 Workspace Volume Index

index, respectively. The variation ranges of  du , d are quite different. So (d )max and (du )max are used to normalize the functions between zero and one. These two

Static equilibrium workspace is defined by the number of points where the end-effector can be placed while all the

348

maximum values can be calculated by performing singleobjective optimization for each function.

Therefore, a hybrid GA-PS approach will be used in this paper to provide much more accurate results.

4.2 Overall Stiffness Index

5.1 Genetic Algorithm

An overall stiffness index (OSI) of the manipulator can be defined as follows:

Genetic Algorithms (GAs) are one of the optimization techniques which work based on the principles of natural evolution. The GAs are appropriate for rapid global search of large and nonlinear spaces [14]. Also, GAs are more useful for the discontinuous problem unlike the conventional gradient-based algorithms. In the GAs, possible solutions are encoded as chromosomes, which form an initial random population. Then, each member of the population is evaluated by a fitness function. After that a selection operator based on the fitness values is applied to the population. The individuals with higher fitness values have more chance to be chosen and to reproduce offspring with the use of other operators such as crossover and mutation to create the next generation. This process will be repeated for a number of generations till the termination criterion is satisfied. This termination is specified by a predefined maximum number of generations or population convergence condition. The best individual of the last generation is the optimal solution to the problem. The parameters for the Matlab’s GA Tool are set as Table 1. The upper bound of γbase and γend are selected to be 60 degree based on the result of [9]. When both γbase and γend approach to 60 degree, the workspace volume of the robot becomes zero and therefore the robot is meaningless.

OSI  OSI  ws1

s  su  ws2 ( s )max ( su )max

(19)

Where the weight parameters ws1 , ws2 (ws1 , ws2 ∈ [0 1] and ws1+ws2 =1) show the proportion of s (GSI), and the proportion of su (GSUI) in the overall stiffness index, respectively. The variation ranges of su , s are quite different. So ( s ) max and ( su ) max are used to normalize the functions between zero and one. These two maximum values can be calculated by performing single-objective optimization for each function. 4.3 Overall Mix of Dexterity and Stiffness Index An overall mixed performance index which is a weighted sum of overall dexterity index (ODI) and overall stiffness index (OSI) can be formed as:

ODSI  ODSI  wd1 ws1

d du  wd 2  (d )max ( du )max

s  su  ws2 ( s )max ( su )max

5.2 Pattern Search

(20)

Pattern search techniques are a class of direct search methods for optimization. Like the genetic algorithm, PS is often useful for global optimization. It is suitable for non-smooth or discontinuous objective functions. PS algorithm searches a set of points called a mesh, around the current point. The mesh is created by adding the current point to a scalar multiple of a set of vectors called a pattern. If the algorithm finds a point within the mesh that improves the objective function, the new point becomes the current point at the next iteration. The search will terminate after a minimum mesh size is reached., a pattern search function as our hybrid function will start from the final solution obtained by GA. The parameters for the Matlab’s PS Tool are set as Table 2.

Where the weight parameters wd1, wd2, ws1, ws2 (wd1, wd2, ws1, ws2 ∈ [0 1] and wd1+wd2 +ws1+ws2 =1) show the proportion of d (GDI), the proportion of  du (GDUI), the proportion of s (GSI) and, the proportion of su (GSUI) in the overall mix of Dexterity and Stiffness index, respectively. The selection of weight parameters depends on whether the designer wants the manipulation to be closer to which of the properties.

5. Optimization Methodology An optimization method should be selected to obtain the optimum values of the design parameters. For these functions, it is not possible to solve this problem analytically. Furthermore, derivative–based methods are not appropriate because of the non-smooth behavior of these objective functions. Therefore, intelligent and direct search methods are suitable such as Genetic Algorithm (GA) or Pattern Search (PS) methods or their combination. The random behavior of GA leads to different solutions for various runs. This problem can be solved by using a PS algorithm started from the GA solution so their combination leads to a unique solution.

Table 1 Genetic algorithm parameters Encoding type Population size Maximum generation Creation function Selection function Mutation function Crossover function Lower bounds on H=[ γbase γend rend/rbase] Upper bounds on H=[ γbase γend rend/ rbase]

349

Real 100 300 uniform stochastic Adaptive Scattered [0o 0o 0] [60o 60o 2]

Table 2 Pattern search parameters Poll Method Complete Poll Search Method Complete Search Initial Mesh size Expansion Factor Contraction Factor Max Iteration

those points within the workspace volume, the specified single and multi-objective functions can be evaluated. For the optimization process, a hybrid GA-PS is performed by using Matlab’s Genetic Algorithm and Direct Search Tool [15].

GPS Positive basis 2N on MADS Positive basis 2N on 1 2 0.5 300

6.1 Results Presentation

The results of the three design parameters H= [ γbase γend rend/rbase] are presented through Tables (3-6). The weight parameters of the five single-objective functions are considered to be equal for the optimization process. Therefore, based on the number of functions used in a multi-objective function (k), all the weight parameters are equal to (1/k). With respect to the results of the Tables (36) we can arrive at the following general conclusions:

6. The Results of Optimization For obtaining the static equilibrium workspace volume and stiffness of the robot, it is also necessary to make these eight assumptions:

1- The results of Tables (3-6) show that for all of the objective functions, the γ angle of BP (γbase) is greater than or approximately equal to the γ angle of MP (γend). 2- The workspace volume index (n) increases by increasing rbase and decreases by increasing gamma angle γbase, γend [7]. Also, for a fixed size of rbase and no moments (Mx=My=Mz=0) just like our case (assumptions 1 and 4), the maximum workspace volume index ( nmax ) is independent of rend/rbase and occurs at γbase=γend=0 for both of the search volumes (Zmax/rbase=1 and Zmax/rbase=2) [7, 8]. This workspace volume index (n) is also added as an additional objective function ( wn n nmax ) to the seven objective functions of the Tables 3 and 4 for the optimization problems. In which nmax and wn are respectively the maximum workspace volume index and its weight parameter. Tables 3 and 4 show the optimal parameters of the robot by adding wn n nmax as an additional objective function ( wn n nmax ) to the seven objective functions in the case of Zmax/rbase=1 and Zmax/rbase=2, respectively. In Tables 3 and 4, the maximum value of all the objective functions nearly occurs at γbase=γend=0, rend/rbase=0.5. Therefore, increasing the range of the robot motion in the Z direction (Zmax) has almost no effect on the optimal parameters for these seven objective functions. 3- It is apparent from Tables 3 and 4 that increasing the range of the robot motion (Zmax) leads to a small reduction in the values of the three optimal parameters γbase , γend ,and rend/rbase for most of the objective functions. 4- The optimal parameters of the robot are presented through Table 5 and 6 without considering the workspace volume index (n) as an additional objective function in the case of Zmax/rbase=1 and Zmax/rbase=2, respectively. It can be observed from Tables 5 and 6 that the values of the optimal parameters obtained by the overall stiffness index OSI are nearly close to the values of the optimal parameters derived by the

1- rbase= 6 m 2- The optimal configuration of the cable robot can be defined as [5].

x  y       0

 base   end ) 2   rend  rbase cos(60  base end ) 2 Therefore, zoptimal is bounded for any values of γbase and zoptimal  rbase sin(60 

as: 0 

zoptimal

 0.8660 . Based on this range, the rbase search volume is considered to be two types with these dimensions: 0≤Z≤Zmax , - rbase ≤Y≤ rbase , - rbase ≤X≤ rbase Where Z max 1  rbase  2

γend

In both of the search volumes, the optimal possible configurations are always inside the two search volumes. 3- The step size along all X, Y and Z axes (  =  x  y  z ) is  0.05 . rbase 4-Fext= [0 0 100 0 0 0] T kN 5-E=200 Gpa 2 6- A=0.0004 m 7- The rotation of MP is equal to zero: ψ=0, θ=0 and ϕ=0. 8-All cables have negligible mass and shape a straight line between the connection points on the base platform and the connection points on the moving platform. A program was created with the use of Matlab software which tests the points within the search space to define if they belong to the static equilibrium workspace volume of the robot. By setting the eight mentioned assumptions, the values of the parameters of the robot, and using Eq. (17), one can obtain this workspace volume of the robot. For

350

Table 3

stiffness uniformity index su . Also, the values of the optimal parameters obtained by the overall dexterity index ODI are nearly close to the values of the optimal parameters derived by the dexterity uniformity index du and finally, the values of the optimal parameters obtained by the overall mix of dexterity and stiffness indices ODSI are nearly close to the values of the optimal parameters derived by the stiffness uniformity index su . 5- It is clear from Tables 5 and 6 that the values of the three optimal parameters obtained by the stiffness uniformity index su are greater than their corresponding values in case of using the stiffness index s . Also, the values of the three optimal parameters obtained by the dexterity uniformity index du are also greater than their corresponding values in case of using the dexterity index d . In addition, the values of the three optimal parameters obtained by the dexterity index d are greater than or equal to their corresponding values in case of using the stiffness index s and finally, the values of the three optimal parameters obtained by the dexterity uniformity index du are greater than their corresponding values in case of using the stiffness uniformity index su . 6- It can be observed from Tables 5 and 6 that the BP is almost a regular hexagon except in the case of stiffness index s and dexterity index d . Also, the MP is always an equilateral triangle in case of stiffness index s and dexterity index d . 7- It is obvious from Tables 5 and 6 that increasing the range of the robot motion (Zmax) leads to a reduction in the values of the three optimal parameters γbase , γend ,and rend/rbase for most of the objective functions. 8- By comparing the results of the Tables 3 and 5 both in case of Zmax/rbase=1, one can find that using the workspace volume index (n) as an additional objective function results in a great reduction in the values of the three optimal parameters γbase , γend ,and rend/rbase for all of the objective functions. 9- Based on the results of the Tables 4 and 6, the same principal is also valid in case of Zmax/rbase=2, one can realize that using the workspace volume index (n) as an additional objective function results in a great reduction in the values of the three optimal parameters γbase , γend ,and rend/rbase for all of the objective functions.

The optimal parameters of the robot by adding wn additional objective function in case of

n nmax

as an

Z max 1 rbase

Objective Function

γbase

rend /rbase

γend

Max( su ) Max(s ) Max(d )

0.8122

0.5062

0.0675

Function Value 0.5711

0.8389

0.5067

0.0566

0.9644

0.8965

0.5069

0.0253

0.9663

Max(du )

0.1875

0.5022

0.4107

0.6439

Max(OSI ) Max(ODI ) Max(ODSI )

0.1562

0.5018

0.4278

0.6901

0.4688

0.5040

0.2578

0.7401

0.7591

0.5059

0.1000

0.6584

Table 4 The optimal parameters of the robot by adding wn additional objective function in case of

n nmax

as an

Z max 2 rbase

Objective Function

γbase

rend/rbase

γend

Max( su )

0.0395

0.5007

0.2740

Function Value 0.6503

Max(s ) Max(d ) Max(du ) Max(OSI ) Max(ODI ) Max(ODSI )

0.0001

0.5002

0.0000

0.9994

0.0000

0.5000

0.0001

0.9999

0.0625

0.5014

0.4787

0.7008

0.0101

0.5000

0.0000

0.7664

0.1767

0.5018

0.2500

0.8001

0.1006

0.5007

0.0000

0.7399

Table 5 The optimal parameters of the robot in case of

351

Z max 1 rbase

Objective Function

γbase

rend/rbase

γend

Max( su ) Max(s ) Max(d ) Max(du ) Max(OSI ) Max(ODI ) Max(ODSI )

59.7689

0.8916

23.5912

Function Value 0.0116

21.1620

0.6505

0.0002

0.1846

39.8159

0.7650

0.0001

0.4166

59.0312

0.8964

32.5931

0.1692

59.4609

0.8735

22.8916

0.8502

59.6464

0.8971

31.9473

0.8663

60.0000

0.8852

23.0902

0.8207

Table 6

[6] Z. Gao, D. Zhang, and Y. Ge, Design optimization of a spatial six degree-of-freedom parallel manipulator based on artificial intelligence approaches, Robotics and ComputerIntegrated Manufacturing, 26(2) , 2010, 180-189. [7] J. Hamedi, H. Zohoor, Kinematic Modelling and Workspace Analysis of a Spatial Cable Suspended Robot as In completely Restrained Positioning Mechanism, International Journal of Mechanical Systems Science and Engineering,2008,109-119. [8] J. Hamedi, A. Bahrami, M. Nikkhah-Bahrami, Workspace sensitivity analysis of spatial cable robots, The IASTED International Conference on Robotics, Thailand , Phuket ,2010,175-182. [9] J. Pusey, A. Fattah, S. Agrawal, and E. Messina, Design and workspace analysis of a 6-6 cable-suspended parallel robot, Mech. Mach. Theory, 39, 2004, 761–778. [10] J. K. Salisbury, and J. J. Craig, Articulated Hands: Force Control and Kinematic Issues, Int. J. Robot. Res. 1(1), 1982, 4-17. [11] C. Gosselin, and J. Angles, A global performance index for the kinematic optimization of robotic manipulators, Trans ASME J. Mech. Des, 113(3), 1991, 220-226. [12] C. Gosselin, Stiffness Mapping for Parallel Manipulators, IEEE Trans. Robot. Autom., 6, 1990, 377–382. [13] B. S. El. Khasawneh, and P. M. Ferreira, Computation of stiffness and stiffness bounds for parallel link manipulators, Int. J. Machine Tools and Manufacture, 39 (2), 1999, 321– 342. [14] J. Yang and V. Honavar, Feature Subset Selection Using a Genetic Algorithm, IEEE Intell. Syst, 13(2), 1998, 44-49. [15] The Mathworks, Matlab’s Genetic Algorithm and Direct Search Toolbox User’s Guide, (Massachusett: The Matworks, 2005).

Z The optimal parameters of the robot in case of max  2 rbase

Objective Function

γbase

rend/rbase

γend

Max( su ) Max(s ) Max(d )

51.9486

0.8294

10.2303

Function Value 0.0055

0.9432

0.5071

0.0001

0.1652

0.9432

0.5071

0.0001

0.3917

Max(du )

59.2928

0.8751

20.0165

0.1215

Max(OSI )

51.9400

0.8293

10.2300

0.8349

Max(ODI ) Max(ODSI )

59.9022

0.8920

19.9193

0.8302

59.8745

0.8714

2.9110

0.8017

7. Conclusion This paper presents the optimal design of a 6-DOF cable robot with respect to multi-objectives based upon the hybrid GA-PS method. Different objective functions such as Global Dexterity Index (GDI), Global Dexterity Uniformity Index (GDUI), Global Stiffness Index (GSI), Global Stiffness Uniformity Index (GSUI), and Workspace Volume Index (n) are considered for the optimization process. Firstly, these indices are used separately to obtain the architectural parameters of the robot. Then, mixed performance indices are specified and optimized to obtain the optimal parameters of the robot such as the size of the moving platform, the size of the fixed platform, and the geometric shapes of the two platforms. The obtained results are valuable in designing these kinds of robots under different conditions. Future work will relate to considering cable length limits and upper tension bounds in the model and optimization process. In the same way, we will include additional objective functions and another optimization method called Multi-Objective Evolution Algorithm (MOEA) to provide a complete set of optimal parameters of the cable robot.

References [1] S. Kawamura, W. Choe, S. Tanaka, and S. Pandian, Development of an ultrahigh speed robot FALCON using wire drive system, in Proc. IEEE/ICRA Int. Conf. Robot. Autom., Nagoya, Japan, 1995, 215–220. [2] J. Albus, R. Bostelman, and N. Dagalakis, The NIST RoboCrane, J. Robot. Syst., 10 (5), 1992, 709–724. [3] L. L. Cone, Skycam: An aerial robotic camera system, Byte, 1985, 122–132. [4] A. Ming, and T. Higuchi, Study on multiple degree-offreedom positioning mechanism using wires (part 1)concept, design and control, International Journal of Japan Social Engineering, 28(2), 1994, 131–138. [5] A. Fattah, and H. A. M. Ghasemi, Isotropic design of spatial parallel manipulators, The International Journal of Robotics Research, 2002, 811-824.

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